the correct size), we have Row-addition transformations The final type of row operation on a matrix adds row multiplied by a scalar to row . The corresponding elementary matrix is the identity matrix but with an in the position. So is the matrix produced from by adding times row to row . And is the matrix produced from by adding times column to column . Coefficient wise, the matrix is defined by : Properties These transformations are a kind of shear mapping, also known as a transvections. The inverse of this matrix is given by The matrix and its inverse are triangular matrices. Therefore, for a square matrix (of the correct size) we have Row-addition transforms satisfy the Steinberg relations. See also Gaussian elimination Linear algebra System of linear equations Matrix (mathematics) LU decomposition Frobenius matrix References Linear algebra